ECE 434: Random Processes Exam 2, University of Illinois at Urbana-Champaign, Spring 2003, Exams of Electrical and Electronics Engineering

The spring 2003 exam 2 for the ece 434: random processes course at the university of illinois at urbana-champaign. The exam covers various topics related to markov processes, brownian motion, poisson processes, and gaussian processes. Students are required to solve problems on finding q matrices, first-order probability distributions, markov properties, martingales, expected values, and probabilities.

Typology: Exams

Pre 2010

Uploaded on 03/16/2009

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University of Illinois at Urbana-Champaign
ECE 434: Random Processes
Spring 2003
Exam 2
Monday, April 14, 2003
Name:
You have 75 minutes for this exam. The exam is closed book and closed note, except that you
may consult both sides of two sheets of notes, typed in font size 10 or equivalent handwriting
size.
Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.
Write your answers in the spaces provided.
Please show all of your work. Answers without appropriate justification will receive
very little credit. If you need extra space, use the back of the previous page.
Score:
1. (9 pts.)
2. (8 pts.)
3. (6 pts.)
4. (8 pts.)
5. (9 pts.)
Total: (40 pts.)
1
pf3
pf4
pf5

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University of Illinois at Urbana-Champaign

ECE 434: Random Processes

Spring 2003 Exam 2

Monday, April 14, 2003

Name:

  • You have 75 minutes for this exam. The exam is closed book and closed note, except that you may consult both sides of two sheets of notes, typed in font size 10 or equivalent handwriting size.
  • Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.
  • Write your answers in the spaces provided.
  • Please show all of your work. Answers without appropriate justification will receive very little credit. If you need extra space, use the back of the previous page.

Score:

  1. (9 pts.)
  2. (8 pts.)
  3. (6 pts.)
  4. (8 pts.)
  5. (9 pts.)

Total: (40 pts.)

Problem 1 (9 points) Let X be a stationary continuous-time Markov process with the transition rate diagram shown.

(a) Write down the Q matrix and find the first-order probability distribution π. (So π is a probability vector, representing the distribution of Xt for each t.)

(b) Let X^2 denote the process X^2 = (X t^2 : t ∈ IR). Is X^2 a Markov process? Justify your answer.

(c) Is (Xt : t ≥ 0) a martingale? Justify your answer.

Problem 3 (6 points) Let (Nt : t ≥ 0) be a Poisson process with rate λ > 0. Find P [N 2 ≥ 3 |N 4 = 4].

Problem 4 (8 points) Let Z = (Zt : t ∈ IR) be a mean zero, stationary Gaussian process with RZ (τ ) = e−|τ^ |^ and let V =

∫ (^) ∞ 0 e

−tZtdt.

(a) Find E[V 2 ].

(b) Find P [V ≥ 3].